#8096296

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(10,14)(15,21)(16,22)(23,24)(25,26,32,31)(27,29,36,33)(28,37,30,34)(35,40,38,39);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,33)(28,38)(29,31)(30,35)(32,36)(34,39)(37,40);(1,3)(2,5)(4,8)(6,7)(25,28,32,30)(26,34,31,37)(27,35,36,38)(29,39,33,40);(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,29)(26,27)(28,39)(30,40)(31,36)(32,33)(34,35)(37,38);(1,3)(2,6)(4,8)(5,7)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,30,32,28)(26,34,31,37)(27,35,36,38)(29,40,33,39);(1,2)(3,5)(4,7)(6,8)(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,31,32,26)(27,29,36,33)(28,37,30,34)(35,39,38,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003492);